The Core Equation: A Density-Position Framework for Central Field Systems

This paper presents the Core Equation, defined in its simple form as H = ρ²x and in its formal form as H = [ρ(ρ₀)]²x. The equation describes the density distribution of a central field: systems in which intensity is greatest at the source and decreases with distance. H is not asserted as a conservation invariant but as a characteristic descriptor of the field — given H and position x, the local density ρ is fully determined. The formal equation introduces the monovariable function ρ(ρ₀), a functional slot controlled by a single parameter ρ₀, which governs density behavior of the medium without altering the core architecture. This property is termed Medium Sensitivity. Applications are explored across volumetric rendering, structural engineering, areal density modeling, and signal propagation.